Non - we find that this represents the sum of two states,...

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Non-normalisable wave functions I have argued that solutions to the time-independent Schrödinger equation must be normalised, in order to have a the total probability for finding a particle of one. This makes sense if we think about describing a single Hydrogen atom, where only a single electron can be found. But if we use an accelerator to send a beam of electrons at a metal surface, this is no longer a requirement: What we wish to describe is the flux of electrons, the number of electrons coming through a given volume element in a given time. Let me first consider solutions to the ``free'' Schrödinger equation, i.e., without potential, as discussed before. They take the form (6.1) Let us investigate the two functions. Remembering that
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Unformatted text preview: we find that this represents the sum of two states, one with momentum , and the other with momentum . The first one describes a beam of particles going to the right, and the other term a beam of particles traveling to the left. Let me concentrate on the first term, that describes a beam of particles going to the right. We need to define a probability current density. Since current is the number of particles times their velocity, a sensible definition is the probability density times the velocity, (6.2) This concept only makes sense for states that are not bound, and thus behave totally different from those I discussed previously....
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